Why are Maxwell's equations and the Lorentz force "so different"

[greypilgrim]

TL;DR Summary

EM fields and matter interact with each other according to Maxwell's equations and the Lorentz force equations. Why are these equations so different; and what happens if you reverse time?

Hi.

Maxwell's equations tell us how charges and currents act on electric and magnetic fields. However, when we conversely want to investigate how EM fields act charges and currents, we need this very different thing called Lorentz force.

This all looks so asymmetric to me because those laws look so different. What happens if I record some kind of interaction between EM fields and matter and then play it backwards to essentially reverse cause and effect? Shouldn't then Maxwell's equation somehow turn into Lorentz force and vice versa?

 

[anuttarasammyak]

Lorentz force formula is derived from integral form of Maxwell equation. e.g. see https://en.wikipedia.org/wiki/Lorentz_force .

 

[vanhees71]

It's much easier if you work with continuous medium, i.e., treat the matter as a charged fluid. Then you can derive the "Lorentz-force density" together with the Maxwell stress tensor from the local momentum conservation of the electromagnetic field + fluid. Everything then gets more symmetric. It becomes particularly beautiful and symmetric, of course, when doing everything relativistically and in manifestly covariant notation.

 

[greypilgrim]

Lorentz force formula is derived from integral form of Maxwell equation. e.g. see https://en.wikipedia.org/wiki/Lorentz_force .

That's interesting. Last week I searched for a derivation of Coulomb's law from Maxwell's equation, and all the answers I found said this is only possible if one uses the Lorentz force equation as an additional assumption.

 

[Vanadium 50]

what happens if you reverse time?

You mean change divB = 0 to 0 = divB?

The reason you need the Lorentz force law to derive Coulomb's Law is because you need to define the E and B fields somehow. Usually its done this way. If you are willing to accept them as postulates, you don't need this and can get there with Gaussian surfaces.

 

[greypilgrim]

Lorentz force formula is derived from integral form of Maxwell equation. e.g. see https://en.wikipedia.org/wiki/Lorentz_force .

Actually, the derivation on Wikipedia seems to at least postulate the electric part $\vec{F}=q\vec{E}$, doesn't it?

The reason you need the Lorentz force law to derive Coulomb's Law is because you need to define the E and B fields somehow. Usually its done this way.

That makes sense. Though if I recall correctly, when I attended an electrodynamics class at university back in the days we started with Maxwell's equations right away and did a lot just manipulating them before the Lorentz force was introduced.

 

[vanhees71]

This is a somewhat unfortunate approach, except if it's a theory lecture assuming you had a good experimental lecture about E&M before. However, as any fundamental concept, also the electromagnetic field (or fields in general) must be somehow operationally introduced. The ideal, logical way were if you introduce special relativity first and then discuss the solution for the problem with the conservation laws (particularly momentum conservation) in a theory, where action-at-a-distance interactions do not work (from relativistic causality). This leads to the conclusion that Faraday's ingenious heuristics is one (and even today the only) solution for this riddle, i.e., the idea of local interactions rather than action-at-a-distance forces.

The most simple way to start E&M thus still is to discuss the em. force on point particles (leaving out the unsolved radiation-reaction problem at this point of course) in terms of the Lorentz force (including from the very beginning both electric and magnetic field components, emphasizing that there's one and only one electromagnetic field, and the distinction between electric and magnetic parts is a frame-dependent concept).

Then, if you also have the action principle in your theoretical toolkit, together with gauge invariance, you can pretty straightforwardly get also to the Maxwell equations for a closed system of fields and charged particles (or at this point better also treat the particles in terms of a continuum theory).

 

[anuttarasammyak]

Actually, the derivation on Wikipedia seems to at least postulate the electric part F→=qE→, doesn't it?

Yea, in my old textbook, in static case electric field is defined by force on test charge particle,i.e.
$$\mathbf{E}:=\lim_{q \rightarrow 0}\frac{\mathbf{F}}{q}$$
Infinitesimal q is introduced so that we can neglect effect of q to the environment generating E.
How E is introduced and defined in your textbook ?